Spectrally-large scale geometry in cotangent bundles
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908988212248576 |
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| author | Feng, Qi Zhang, Jun |
| author_facet | Feng, Qi Zhang, Jun |
| contents | In this paper, we prove that the ${\rm Ham}$-orbit space from a fiber of a large family of cotangent bundles, as a metric space with respect to the Floer-theoretic spectral metric, contains a quasi-isometric embedding of an infinite-dimensional normed vector space. The same conclusion holds for the group of compactly supported Hamiltonian diffeomorphisms of some cotangent bundles. To prove this, we generalize a result, relating boundary depth and spectral norm for closed symplectic manifolds in Kislev-Shelukhin's recent work, to Liouville domains. Then we modify Usher's constructions (which were used to obtain Hofer-large scale geometric properties) to achieve our desired conclusions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17590 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectrally-large scale geometry in cotangent bundles Feng, Qi Zhang, Jun Symplectic Geometry Metric Geometry 53D40, 53D22 In this paper, we prove that the ${\rm Ham}$-orbit space from a fiber of a large family of cotangent bundles, as a metric space with respect to the Floer-theoretic spectral metric, contains a quasi-isometric embedding of an infinite-dimensional normed vector space. The same conclusion holds for the group of compactly supported Hamiltonian diffeomorphisms of some cotangent bundles. To prove this, we generalize a result, relating boundary depth and spectral norm for closed symplectic manifolds in Kislev-Shelukhin's recent work, to Liouville domains. Then we modify Usher's constructions (which were used to obtain Hofer-large scale geometric properties) to achieve our desired conclusions. |
| title | Spectrally-large scale geometry in cotangent bundles |
| topic | Symplectic Geometry Metric Geometry 53D40, 53D22 |
| url | https://arxiv.org/abs/2401.17590 |